Let \( X_1, X_2, \dots, X_n, \dots \) be in­de­pend­ent ran­dom vari­ables and let \( S_n = \sum_{\nu=1}^n X_{\nu} \). In the so-called law of the it­er­ated log­ar­ithm, com­pletely solved by Feller re­cently, the up­per lim­it of \( S_n \) as \( n\to\infty \) is con­sidered and its true or­der of mag­nitude is found with prob­ab­il­ity one. A coun­ter­part to that prob­lem is to con­sider the lower lim­it of \( S_n \) as \( n\to\infty \) and to make a state­ment about its or­der of mag­nitude with prob­ab­il­ity one.

The lim­it­ing dis­tri­bu­tion of the max­im­um cu­mu­lat­ive sum of a se­quence of in­de­pend­ent ran­dom vari­ables has been dis­cussed re­cently by Er­dős–Kac [1946] and Wald [1947]. Er­dős and Kac treated the case where each ran­dom vari­ables has zero mean, while Wald con­sidered more gen­er­al cases. We shall show that the prob­lem can be treated by a uni­form meth­od start­ing with a clas­sic­al com­bin­at­or­i­al for­mula due to De Moivre.

In the clas­sic­al coint-toss­ing game we have a se­quence of in­de­pend­ent ran­dom vari­ables \( X_{\nu} \), \( \nu=1,2,\dots \), each tak­ing the val­ues \( \pm 1 \) with prob­ab­il­ity \( 1/2 \). We are in­ter­ested in the signs of the par­tial sums \( S_n = \sum_{\nu=1}^n X_{\nu} \).

We con­sider a se­quence of in­de­pend­ent ran­dom vari­ables hav­ing the com­mon dis­tri­bu­tion func­tion \( F(x) \) which is as­sumed to be con­tinu­ous. Let \( nF_n(x) \) de­note the num­ber of ran­dom vari­ables among the first \( n \) of the se­quence whose val­ues do not ex­ceed \( x \). Write
\[ d_n = \sup_{-\infty < x < \infty}|n(F_n(x)-F(x))| .\]
Kolmogoroff [1933] proved that the prob­ab­il­ity
\begin{equation*}\tag{1}
P(d_n\leq\lambda n^{1/2}),
\end{equation*}
where \( \lambda \) is a pos­it­ive con­stant, tends as \( n\to\infty \) uni­formly in \( \lambda \) to the lim­it­ing dis­tri­bu­tion
\begin{equation*}\tag{2}
\Phi(\lambda)=\sum_{-\infty}^{\infty}(-1)^j e^{-2j^2\lambda^2}.
\end{equation*}
Smirnoff [1939] ex­ten­ded this res­ult and re­cently [Feller 1948] has giv­en new proofs of these the­or­ems. In this pa­per we shall ob­tain an es­tim­ate of the dif­fer­ence between (1) and (2) as a func­tion of \( n \), val­id not only for \( \lambda \) equal to a con­stant but also for \( \lambda \) equal to a func­tion \( \lambda(n) \) of \( n \) which does not grow too fast.

One as­pect of the the­ory of ad­di­tion of in­de­pend­ent ran­dom vari­ables is the fre­quency with which the par­tial sums change sign. In­vest­ig­a­tions of this nature were ori­gin­ated by Paul Lévy, in a pa­per [1939] which con­tains a wealth of ideas. This prob­lem as such was men­tioned by Feller in his 1945 ad­dress. In the case where the par­tial sums can ac­tu­ally van­ish the prob­lem falls un­der the head of “re­cur­rent events,” a gen­er­al the­ory of which was re­cently de­veloped in a pa­per by Feller [1949]. A very spe­cial case had been stud­ied in de­tail by Hunt and my­self [Chung and Hunt 1949]. Gen­er­al­iz­ing the prob­lem in a nat­ur­al way we shall con­sider the num­ber of times \( T_n \) with which the se­quence of re­duced par­tial sums \( S_k-E(S_k) \), \( k=1,2,\dots, n \) crosses a giv­en value \( c \). We shall es­tab­lish the lim­it­ing dis­tri­bu­tion of \( T_n \) in the case where the ran­dom
vari­ables have a com­mon dis­trib­utino with a fi­nite third ab­so­lute mo­ment.

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